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\author{Francisco Aspillaga \and Alejandro F. Mac Cawley \and Sergio Maturana}
\title{A Bulb Acquisition Planning Model for a Chilean Lily Flower Enterprise}
\doublespacing
\begin{document}
\maketitle
\begin{abstract}
The acquisition process of raw materials in the flower industry, involves a large number of technical, biological and economic variables along with a considerable level of capital involved. For this reason in the present paper we develop a Bulb Acquisition Planning (BAP) mixed integer optimization model. The model incorporates aspects such as substitution, precocity, risk diversification, seasonality, inventory among others. We implemented the model using enterprise data from production planning. Finally the results of the BAP model and the executed plan are compared with respect to cost, container programming and the inventory maintenance.

\smallskip
\noindent \textbf{Keywords.} Aggregate production planning; mixed integer linear programming; bulb acquisition\
\end{abstract}

\section{Introduction}

The world floriculture  market has significantly grown in the past 15 years, from USD 8.5 billion in 2001 to USD 20.6 billions on 2013 \citep{Rabo_2015}. Cut flowers has been the main group within the global floriculture trade, where the exchange of products between countries has become a common practice. As part of this exchange, the countries located in the southern hemisphere, present a competitive advantage of having counter season production with his principal markets such as the United States, Europe and Japan. This advantage enables the countries located in this hemisphere to get to the consumer with a high quality product in a period where higher international prices are registered. 

Chile still has a small share of the international cut flower market with a 0,03\%, with the US being the main market with 82,8\% of the exports to this market \citep{Odepa_2014}. Nevertheless, Chile presents a number of other advantages for the flower production, where we can mention the agro climatic characteristics, which vary enormously along the long extension of the country (From parallel 24 to 55) allowing a large number of species to be grown. The Mediterranean weather, with cold winters and moderates temperatures during the summer season, gives excellent conditions for the development of bulb species. And finally, the presence of a mountain range (Andes Range) to the east, a dessert (Atacama) to the north and the sea (Pacific) to the west; which structures a strong  fitosanitary barrier to the entrance of pests. All of this factors allow Chile to export to markets with high quality standards. Chile is the fourth exporting country in the world of flower bulbs, with a 1,8\% share of the international bulb flower market and its considered to be an attractive country for its  production \citep{Odepa_m_2014}

Because of the large distance of Chile to the principal sources (Netherlands) and buyers markets of the bulbs is that the transportation costs have a direct impact in the production costs of the flowers. Since most of the best quality genetic material is imported, with high costs per unit, the bulbs generally comprises about 48\% of gross margins \citep{Kang01102013}. Because of the high costs involved in the acquisition and transportation of the vegetal material, is that a decision and support planning system would be required to aid the administration and assignment of resources in the acquisition of bulbs.

Therefore, the objective of this work is to describe and analyze the bulb acquisition planning process of a lily flower enterprise, in order to describe the variables that influence the decision process from a mathematical stand point, which will finally allow it to be optimized and support the planning process. To accomplish this objective, a mixed integer mathematical programming model was developed, the objective function is to minimize the relevant costs incurred in the acquisition process subject to a series of restrictions. The model was validated with data from the local flower industry and  the results where compared with the cost data obtained from the executed production plan.

The contribution of this paper is to acknowledge and expose the implicit factors that reigns the decision process of the planning and acquisition of the bulbs in a flower enterprise; incorporate and integrate this factors in a mathematical long term planning model to obtain an optimal answer and finally; contribute with an operational tool which can support the planning process of lily flower enterprise, reducing the time and effort given to this process.

\section{Definition of the Bulb Acquisition Planning Problem and Variables}

\subsection{General Characteristics}

Every lineal optimization model incorporates three basic element \citet{dent1986}: (1) An objective function, which minimizes or maximizes a function with a set of levels of activities; (2) a description of the activities of the system, through the coefficients that represents the productive responses of the system; and (3) a set of restrictions which defines the operational conditions and limits of the productive system. A large number of authors have proposed a diverse number of models and algorithms for the description and optimization of production planning problems \citet{ahumada2009} and their applications, in this area we can cite the works of \citet{holt1955}; \citet{heady1954}; \citet{fokkens1981}; \citet{hax1979}. 

Some more works in the area of production planning are \citet{nam1992} where they review models and methodologies in aggregate production planning. \citet{billington1983} and  \citet{maes1988} does a full revision of the lot sizing problem \citet{rajagopalan2001}.
In the application of production planning models in agriculture, Glen \citet{glen1987}, describes an extensive review of mathematical models applied to agricultural planning. Applications of production planning in the forest sector con be seen in the papers of  \citet{weintraub1996};  \citet{laroze2000};  \citet{palma2001}. Finally, \citet{broekmeulen2002} and \citet{caixeta2002} presents mathematical models applied in the flower industry. The first, does a comparative analysis of three different optimization algorithms for the treatment of bulbs, in order to minimize the cost of batch movement during treatment. The second, implements a production planning system, based in lineal optimization which objective is to maximize the total contribution margin of a flower production company.

In the present paper, we propose a Bulb Acquisition Planning Model (BAP), which objective is to find the best combination of resources necessary to fulfill the market demand of flowers. The model allows the acquisition manager to find the most suitable combination of varieties, which allows him to reach the least cost. The biological material is acquired from international suppliers in different periods during the year. It also optimizes the shipment organization, the quantity and type of bulbs to be acquire, programming the quantity of inventory to be held by the company.

In order to accomplish the objectives above described, the 	model minimizes the total acquisition costs involved in the process, which consist of the costs of the vegetable material to be acquired, the transport costs and the costs associated to the inventory. The planning horizon for the model is one year, because the acquisition and negotiation process is done once in the year with the suppliers. The modeling unit is the bulb due to the fact that this is minimum unit used by the decision maker.

	For a better understanding of the problem that the acquisition manager faces, we will proceed to do an explanation of the variables and the decision process that governs the acquisition process. 

\subsection{The Bulb Acquisition Process}

In the figure 1 we can observe the structure of the actual bulb acquisition process. The first and more important parameter to be inputted is the flower market demand forecast; with this information the decision maker explodes the whole acquisition plan. In first place, he analyses the total flower demand of the company and the costs of the bulbs to be acquired, then he studies the feasibility to substitute some of this demand with lower cost biological material, depending on the characteristics of the market and the material. Once he has supplied the whole flower demand, he analyses the level of risk diversification in the acquisition of material, diversification stated as the percentage of the material supplied by only one company. This diversification is used to lower the risk of defective biological material provided from one supplier, so he uses different suppliers for a single variety. The level or percentage of diversification will depend on the characteristics of the supplier.

Figure 1: Structure of the Bulb Acquisition Process 
 

In stage II, the decision maker contrasts the elaborated plan with the availability of the vegetal material to be acquired, if the material is available he goes to stage III else he has to substitute a larger portion of the material or not fulfill the whole demand.

	In stage III he takes the plan which availability has been checked and analyses the precocity and season availability of the varieties he has chosen. If the precocity or seasonal availability is not covered he has to find a new substitute for the variety.
	
	Stage IV, he structures the shipment organization. The rule the decision maker uses is to: minimize the number of shipments subject to arrival on time of the bulbs in order to plant them.
	
	Finally in stage V, the acquisition manager presents the plan to the CEO in order to be approved and implemented.
	
\subsection{Definition of Costs related to the Bulb Acquisition Process}

The costs related to the bulb acquisition process can be defined by:

-	Acquisition Costs: Biological material acquired; values FOB or Ex works which depends upon the commercial agreement between supplier and buyer.
-	Transport Costs: The freight, insurance and transport costs between port and production facility. Warehouse costs, if needed, are also included.
-	Inventory Costs: correspond to the immobilized capital due to the acquisition of the material.

The sum of these three costs adds the total cost associated to the acquisition of the vegetal material.

The BAP model is defined as a minimization of the acquisition, transport and inventory costs. Subject to a whole series of technical restrictions, which are: maximum substitution of varieties, production precocity of the varieties, risk diversification, season and availability of the biological material. The model incorporates restrictions which allows the demand to be fulfilled, the minimum time required for the bulb to develop into a flower and the plantation to be done according to the acquisition of the bulbs. In the process of substitution of varieties the model has imbibed restrictions which allow a maximum level of substitution of individual varieties and the availability of them. The enterprise has a policy about  risk of defective biological material with suppliers, this policy is that a maximum percentage of the lot of biological material can be acquired from certain suppliers and a minimum has to be acquired from certain certified suppliers.

\section{Model}

The following nomenclature was used in the mathematical formulation:

\subsubsection{Indexes}

$i_D$ = Variety ($1,..,n$) demanded.

$i_A$ = Variety ($1,..,n$) acquired.

$j$ = caliber ($1,..,m$).

$l$ = bud ($1,..,q$).

$k$ = supplier ($1,..,r$).

$t$ = time ($1,..,t$), t is the date in which the batch arrives the production center.

$t_F$ =  time ($1,..,t$) to flower.

$b$ = geographical origin of the bulbs eg. Chile or Netherlands(1,..,y).

\subsubsection{Parameters}

$CMV_{i_Ajk}$ =  Cost of the variety acquired $i_A$, of caliber $j$, obtained from supplier $k$.

$CT_b$ = Cost of transport from origin $b$ (terrestrial o maritime) CIF price.

$MCmin_{i_Ak}$ = Minimum quantity of bulbs matrix of the variety $i_A$ assigned to supplier $k$ as a percentage of the total demand.

$CAP_b$ = Maximum capacity of trays that a container can be filled from origin $b$.

$MO_{kb}$ = Binary matrix that assigns supplier $k$ to origin $b$.

$PB_{i_Aj}$ = Maximum quantity of bulbs of variety $i_A$ and caliber $j$ that a tray may contain.

$DISP_{i_Ajk}$ = Availability of variety $i_A$, of caliber $j$ from supplier $k$.

$MS_{i_Ai_D}$ = Binary matrix of feasibility of substitution of demanded variety $i_D$ by acquired variety $i_A$.

$D_{i_Dt_fl}$ = Market demand of variety $i_D$, in flower time $t_F$, and bud $l$.

$MBC_{i_Djl}$ = Conversion matrix of bud $l$ and caliber $j$, of the demanded variety $i_D$.

$MST_{i_D}$ = Maximum convertibility matrix of the demanded variety $i_D$.

$E_{tb}$ = Binary matrix of the seasonal availability in time $t$ from the origin $b$.

$Inv0_{i_Aj}$ = Initial inventory of variety $i_A$ of caliber $j$.

$Ef$ = Efficiency of production.

$r$ = Interest rate.

$MP_{iDtt_f}$ = Precocity binary matrix which relates the arrival time $t$ and the harvest time of the flower $t_F$, for each demanded variety $i_D$.

\subsubsection{Decision variables.}

$X_{i_ajktb}$ = Amount of bulbs of variety $i_A$ and caliber  $j$, to be obtained from supplier $k$, in time $t$, from origin $b$.

$B_{tb}$ = Batch integer variable to be transported from origin $b$ in time $t$.

$SUS_{i_Ai_Djtt_F}$ = Amount of demanded variety $i_D$ to be substituted by variety $i_A$, of caliber $j$, to be acquired in time $t$ and harvested in flower time $t_F$.

$NS_{i_Djtt_F}$ =  Amount of demanded variety $i_D$ not substituted, of caliber $j$, to be acquired in time $t$ and harvested in flower time $t_F$.

$Inv_{i_Ajt}$ = Inventory of acquired variety $i_A$, of caliber $j$, in time $t$.

$Pt_{i_Ajtt_F}$ = Planted bulbs of acquired variety $i_A$, caliber $j$, in time $t$, to be harvested in flower time $t_F$.

 The parameter $MP_{i_Dtt_F}$ is a binary matrix, which takes the value of 1 if the precocity of the variety allows the feasible combination of time $(t, t_F)$ which assures the necessary development in order to obtain a commercial flower in  $t_F$. Else takes the value of 0. For all solutions in the problem belong to the set of precocities defined in the matrix $MP$, so $F = \left\lbrace x \mid x \in \mathbb{N}_{+}, MP = 1 \right\rbrace $. All other combination belonging to the complement $F_ c = \left\lbrace x \mid x \in \mathbb{N}_{+}, MP = 0\right\rbrace $, are unfeasible solutions who escapes the combination $(t, t_F)$ required by the precocity of the varieties. This defined set restrains the universe of feasible solutions to the BAP problem.

\subsection{Objective function}

The objective function is as follows:

\begin{equation}
\min \underbrace{\sum\limits_{i_A,j,k,t \in F,b} X_{i_Ajktb}CMV_{i_ajk}}_\text{Acquisition costs} + \underbrace{\sum\limits_{t \in F,b} B_{tb}CT_b}_\text{Transport costs} + \underbrace{\sum\limits_{i_A,j,k,t \in F} Inv_{i_Ajt}CMV_{i_ajk}r}_\text{Inventory costs}
\end{equation} 

The problem in hand, consists in finding the optimal solution which minimizes the total costs involved in the process. The cost function consists of the costs associated  in the acquisition of the vegetal material, the transport costs and the inventory costs. This function has continuous variables  such as  $X_{i_Ajktb}$, $Inv_{i_Ajt}$ and an integer variable $B_{tb}$  and cost parameters involved in the planning such as  $CMV_{i_ajk}$, $CT_b$ and $r$. The inventory costs has been set to the capital interest rate given by local banks to the company, in order to incorporate the cost of capital through time. 

\subsection{Restrictions}

In order to obtain the required number of bulbs to be acquired from the suppliers, the structure of shipments and the required inventory to fulfill the market demand of flowers, a number of restrictions where applied to the model. The coefficients associated to this restrictions where obtained through two different sources of information: the suppliers and the company. Information of availability of bulbs, shipment capacity, substitution matrix, precocity of the different zones of production, diversification rate in the suppliers, caliber to bud conversion, market demand and other parameters, where obtained from this sources.


The restrictions imposed to the model are as follows:

1.	The fulfillment of the market demand in every time $t_F$. 
\begin{equation}
      \sum\limits_{j,t \in F} \left[ \left( \sum\limits_{i_A} SUS_{i_Ai_Djtt_F} + NS_{i_D jtt_F} \right) MBC_{i_Djl} \right] \geq D_{i_Dt_fl}Ef \qquad \forall i_Dt_fl
\end{equation}                        

2.	The maximum substitutability of bulbs
In this restriction we incorporates the ability to substitute partially a variety of flower for another. Substitution is done in function of the color of the flower in the same specie. This restrictions limits the total percentage of substitution in a given specie (R.3.3.2) and in the total demanded flowers (R.3.3.3).
    \begin{equation}
      \sum\limits_{j,t \in F, t_f \in F}SUS_{i_AiDjtt_F}MBC_{i_Djl} \leq \sum\limits_{t_F \in F}D_{i_Dt_Fl}MS_{i_Ai_D}Ef \qquad \forall i_Ai_Dl
\end{equation}
\begin{equation}      
       \sum\limits_{i_A,j,t \in F}SUS_{i_AiDjtt_F}MBC_{i_Djl} \leq D_{i_Dt_Fl}MST_{i_D}Ef \qquad \forall i_Dt_Fl     
  \end{equation}          
           
3.	Acquired bulbs in time $t$ correspond to the planted bulbs in $t$ and they correspond to a flower in time $t_F$. The combination $(t,tF) \in F$.
\begin{equation}
Pt_{i_Ajtt_F} = \sum\limits_{i_D}SUS_{i_Ai_Djtt_F} + NS_{i_Ajtt_F} \qquad \forall i_Aj(t,tF) \in F
\end{equation}

4. Balance equation: keeps the inter-temporal balance between the inventory for all $t$.	

\begin{equation}
\sum\limits_{k,b} X_{i_Ajktb} + Inv_{i_Aj(t-1)} = Inv_{i_Ajt} + \sum\limits_{t_F} Pt_{iAjtt_F} \qquad \forall i_A,j,(t \neq \min{\{t\}})
\end{equation}

Equation XXX defines the starting inventory as a fixed parameter which is given by the available plant material from the previous year.

\begin{equation}
Inv_{i_Ajz}=Inv0_{i_Aj} \qquad \forall i_a,j,(z= \min{\{t\}})
\end{equation}

Equation XXX forbids the planning model from planting outside of the time $F$.

\begin{equation}
Pt_{i_Att_Fj}=0 \qquad \forall i_A,j,(t,t_F) \notin F
\end{equation}

5.	The acquired number of bulbs have to be less or equal than the current availability of the market and the different providers.

\begin{equation}
\sum\limits_{t,b} X_{i_Ajktb} \leq DISP_{i_Ajk} \qquad \forall i_A,j,k
\end{equation}

6.	The decision makers assign a minimum percentage of the demand to be assigned to an specific provider. By setting a minimum percentage to be purchase to each provider the company look at diversifying the risk of purchasing an entire defective lot from a single provider. If such restriction was not enforce, the model would acquire all of the production from the least cost provider, and since all of the material is sourced from a single provider, there is an increased risk of the batch being defective. This allows the risk to be spread in a group of providers.

\begin{equation}
\sum\limits_{j,b} X_{i_Ajktb} \geq MCmin_{i_Ak}\sum\limits_{j,h,b} X_{i_Ajhtb} \qquad \forall i_A,k,t
\end{equation}

The minimum assigned percentage to each provider is set according to the following parameters: quality of the delivered genetic material (\% of emergence, vigour y lot uniformity), dependability, payment conditions, origin and strategic relations. 

7.	The acquisition of bulbs is related to the shipment from different destinations, given by the integer variable $B_tb$. Also the amount to be shipped depends on the capacity of the container $CAP_b$.

\begin{equation}
\sum\limits_{i_A,j,k} X_{i_Ajktb}PB_{i_Aj} \leq \sum\limits_{k} CAP_{b}B_{tb} \qquad \forall t,b
\end{equation}

8.	The availability of genetic material from the different suppliers is dependent upon the seasonality of the different origins.

\begin{equation}
\sum\limits_{i_A,j,k} X_{i_Ajktb} \leq E_{tb}M_{1} \qquad \forall t,b
\end{equation}

The value of $M_1$ is defined by $M_1=(\sum\limits_{i_D,t_f,l} D_{i_Dt_fl}Ef)$ which corresponds to the maximum possible demand. The value es set as tight as possible to improve the solution speed.

9.	All bulbs provided from a given supplier must come from the same origin.

\begin{equation}
\sum\limits_{i_A,j,t} X_{i_Ajktb} \leq MO_{kb}M_{2} \qquad \forall k,b
\end{equation}

$M_2$ is defined by the following expression $M_2=(\sum\limits_{i_D,t_f,l} D_{i_Dt_fl}Ef)$ as the previous constrain, the values has set as tight as possible to improve solution speed.

10.	Non negativity constraints.

\begin{equation}
X_{i_Ajktb} \geq 0; B_{tb} \in{0,1};SUS_{i_Ai_Djtt_F} \geq 0; NS_{i_Ajtt_F} \ geq 0; Inv_{i_Ajt} \geq 0;  Pt_{iAjtt_F} \geq 0
\end{equation}

The number of varieties and precocity within each species, along with the number of species used and substitution policies, makes the bulbs acquisition process with their respective optimal shipment planning, a very complex task. The BAP includes a large number of integer variables related to the shipping process, just this problems relates to the family of problems know as the "integer Knapsack", which according to \citet{garey1979} corresponds ta NP-complete problem.

\section{Implementation}

The Bulb Acquisition Planning model (BAP) was formulated in the algebraic modeling language AMPL, described by \citet{fourer1993}. The model considered: 54 varieties of Lilium spp., four bulb calibers, four blooming classifications, five suppliers, 78 weeks for the arrival of the plant material, 52 weeks to flower time (demand) and two places of origin of the plant material (the Netherlands and Chile). The problem was composed of 100,581 variables, of which 156 are integer variables and 100,425 are linear variables and 44,669 constraints, all linear. The model was implemented using distributed NEOS optimization server version 4.0 (Czyzyk, 1998) and mathematical programming software XPRESS-MP (Dash Associates, 1999). The run-time of the model was between 1.5 to 2.5 hours. 

\section{Results}

In this section we will present the implementation of the BAP model and how it can be used for decision making at the tactical level of a Chilean flower producing company.

The results of the BAP are presented in Figure\ref{results} were they are compared with the implemented production plan. The data used in the model was composed of real prices adjusted to the exchange rate for each period, risk policies and substitution scheme used by the CEO of the partner company in the actual programming. With production we obtained data regarding the precocity of the varieties and the blooming calibers. The demand for each type of flower is provided by an expert advisor for the company and its for a period of one calendar year. There is a quite extended planning period because the acquisition decisions are made months in advance to obtain an export cut flowers, also we need to consider the time required for ocean freight, the time requires for planting, the precocity the different varieties, among processes that are needed to obtain a final cut flower. Results reflect cost minimization in all items related to the acquisition of the plant material.

\begin{table}[htbp]
  \centering
  \caption{Cost structure comparison between BAP model and the executed yearly production plan.}
    \begin{tabular}{lrr}
    \hline
    Cost component & Executed & BAP \bigstrut\\
    \hline
    \hline
    Toal Cost & 100\% & 97.85\% \bigstrut[t]\\
    Vegetal material cost & 100\% & 99.59\% \\
    Transport cost & 100\% & 103.33\% \\
    Invenotry cost & 100\% & 36.34\% \bigstrut[b]\\
    \hline
    \hline
    \end{tabular}%
  \label{tab:addlabel}%
\end{table}%


\begin{figure}[htp] \centering{
\fbox{\includegraphics[scale=0.45]{figure/figure_1.pdf}}}
\caption{Comparison of the weekly container schedule and the number of bulbs in inventory obtained from the BAP model with the executed plan. Numbers below each container indicate the number of containers.}
\label{results}
\end{figure}

We proceeded to compare the total costs of the transaction, the acquisition costs of the plant material , transportation costs and inventory costs between the actual programming and programming model optimized by the PAB . The results are shown in Table 1 , which shows a comparison of both scenarios in terms of percentage difference.

 The model PAB gave the distribution of shipments optimized for the planning horizon. On the other hand , shows the actual distribution of shipments that took place for the 2002 season. Both results are graphed in the lower part of Figure 1, by way of comparison . The points on the graph indicate the chronological time of arrival of each shipment, and under item number indicates the number of containers in each arrival . Besides, the maintenance policy optimized inventory for each array in relation to the storage time of the plant material. The evolution of the total inventory optimized throughout the year , compared to the actual inventory maintenance for 2002, what is presented in the top of Figure 1.


The delivery model explicitly PAB replacement detail, specifying the pairs of varieties and quantities replaced bulb number. Finally, it also delivers the plans to the amounts of planting, assigning different varieties for planting at a specific time to meet the demand.

En esta sección se muestra con un ejemplo como el modelo PAB puede ser utilizado para la toma de decisiones a nivel táctico de una empresa de flores.
	
Los resultados son presentados en base a la programación de datos correspondiente a la planificación de la temporada 2002 de producción. Los datos incorporaron precios reales ajustados al tipo de cambio correspondiente a cada período y las políticas de riesgo y sustituibilidad utilizadas por el CEO de la empresa colaboradora en la programación real. Los datos también incorporaron las distintas precocidades entre las variedades y la conversión de calibres a botonajes utilizados por el COO en la planificación táctica. La demanda se ingresa como un input aportado por un asesor experto para el período de un año calendario. Este punto es de extrema importancia debido a que las decisiones de adquisición de bulbos se toman con meses de anticipación a la obtención de una vara floral exportable, ya que se deben considerar los tiempos requeridos para el flete marítimo, las faenas de plantación, las precocidades de la diferentes variedades, etc. Los resultados reflejan la minimización de costos en todos los ítem relacionados con la adquisición del material vegetal.

Los resultados obtenidos con el modelo PAB fueron comparados con la programación real de adquisición de bulbos realizada por la empresa colaboradora para el año 2002. Se procedió a comparar los costos totales de la operación, los costos de adquisición del material vegetal, los costos de transporte y los costos de inventario entre la programación real y la programación optimizada por el modelo PAB. Los resultados se muestran en la Tabla  1, donde se aprecia un comparativo de ambos escenarios en términos de diferencia porcentual.

 El modelo PAB entregó la distribución de los embarques optimizada, correspondiente al horizonte de planificación. Por su parte, se presenta la distribución real de embarques que tuvo lugar para la temporada 2002. Ambos resultados se grafican en la parte inferior de la Fig. 1, a modo de comparación. Los puntos en el gráfico indican el momento cronológico de la llegada de cada embarque, y el número bajo el punto, indica la cantidad de containers en cada arribo. Además, se obtuvieron las políticas de mantención de inventario optimizadas para cada variedad en relación al tiempo de almacenaje del material vegetal. La evolución del inventario total optimizado a lo largo del año, se comparó con la mantención de inventario real para año 2002, lo que se presenta en la parte superior de la Fig. 1. 


El modelo PAB entrega explícitamente el detalle de sustitución, especificando los pares de variedades sustituidas y las cantidades en número de bulbos. Por último, también entrega los planos con las cantidades de plantación, asignando las distintas variedades a ser plantadas en un tiempo especifico para satisfacer las demanda. 

\section{Discussion}

El modelo PAB está definido como una minimización de los costos de adquisición, transporte e inventario del material vegetal. El modelo incorpora conceptos relevantes en la toma de decisiones como; costos, correlación entre botonaje y calibre, sustituibilidad de variedades, precocidades de producción, mantención y costo de inventario, diversificación de riesgo, centros de origen, estacionalidad de la disponibilidad del material vegetal y costo de capital. El modelo PAB integra todos estos conceptos y entrega una solución que incorpora toda la gama de posibilidades, dejando de manera explicita la interrelación entre ellas y otorgando una solución óptima que cumpla con los requerimientos de demanda. Por su parte, el tomador de decisiones particiona el proceso  en varias etapas. Primero, procede a seleccionar el material vegetal a adquirir en base al costo de los bulbos, según las cotizaciones entregadas por los diferentes proveedores, las posibilidades de sustitución entre variedades y la diversificación del riesgo. Luego, comprueba la precocidad de las variedades adquiridas y la estacionalidad de la disponibilidad de cada proveedor. Por último, organiza la programación de los embarques, ajustándose al cumplimiento de la demanda y a las dos primeras etapas. Así, al comparar los resultados entre ambas metodologías (programación optimizada y programación real), no se observa una diferencia significativa en el costo de adquisición del material vegetal, lo que se traduce en que el tomador de decisiones ya estaba logrando la optimalidad en la selección de las variedades.

Sin embargo, sí se observó una gran diferencia en cuanto al costo de inventario. La programación optimizada entregó un ahorro en los costos de mantención de inventario del 63.66\% con respecto a la programación real. Esto fue posible mediante una adecuada distribución en la programación y la cantidad de containers adquiridos. Lo anterior provoca que los costos de transporte se eleven un 3.33\% en comparación a la programación real, lo que se justifica plenamente si se considera el ahorro porcentual en los costos de mantención de inventario. 
En términos monetarios, se obtuvo un ahorro total de US\$ 40.000 mediante la utilización del modelo PAB, en relación a la programación real. Este ahorro se enmarca en el horizonte de planificación de un año calendario. El modelo PAB entrega una distribución de inventario mejorada, la que busca disminuir al mínimo la cantidad de inventario e incorporar el costo del capital invertido. Esto se refleja en que el 85\% del ahorro total, correspondió a ahorros en este ítem.

La programación obtenida con el modelo PAB, fue cotejada con los ejecutivos de la empresa colaboradora. Se verificó que la adquisición cumpliera con los requisitos de sustituibilidad máxima, existiera concordancia entre los botonajes demandados y los calibres de bulbos adquiridos, se cumpliera con la demanda y se respetaran las políticas de diversificación de riesgo. La empresa colaboradora otorgó su aceptación en base a la evidencia de que los resultados entregados por el modelo se comportaron de manera fiel y confiable.

La contribución de este trabajo se centra en tres ejes principales. En primer lugar, contribuye a aclarar y exponer los factores implícitos que gobiernan la decisión al momento de adquirir el material vegetal en una empresa inmersa en la industria de la floricultura. En segundo lugar, el modelo integra todos los elementos que conforman la decisión para entregar una solución óptima y muestra de manera explícita los diferentes costos incurridos en la operación. Esto genera un plano multidimensional que hace difícil que el tomador de decisiones lo visualice por si solo. Además, incluye y deja claramente de manifiesto el costo de capital por concepto de capital inmovilizado en inventario. En tercer lugar, el modelo constituye una herramienta operacional de gran importancia. Permitirá a los ejecutivos de la empresa obtener la planificación de adquisición de bulbos y la programación de embarques en un corto período de tiempo, integrando todas las variables relevantes y asociado directamente los bulbos adquiridos a las plantaciones y a su mantención en inventario.

Finalmente, el modelo PAB se caracteriza por ser un modelo de tipo determinístico, ya que no considera cambios en la demanda, lo que podría generar distintos escenarios que escapen a la solución entregada. Este tema abre una puerta para futuras investigaciones mediante el uso de modelos de programación entera mixta estocásticos. En cuanto a los tiempos de resolución, se estima necesario generar a futuro algoritmos de solución que permitan acelerar el proceso de modelo y mejore su desempeño como herramienta de negociación. 

\section{Conclusion}

	En este paper nosotros exploramos en el desarrollo e implementación de un modelo de planificación agregada de la producción orientado a los procesos de adquisición del material vegetal en una empresa de flores en la Zona Central de Chile. El objetivo central de este paper fue abordar un sistema productivo agrícola, desde un punto de vista holístico que incorpora aspectos técnicos, biológicos, económicos y estratégicos. Así, se buscó mejorar la planificación en la distintas etapas, mediante el uso de herramientas de optimización, que permitieran abordar el sistema agrícola desde un punto de vista matemático-cuantitativo.
	Nosotros presentamos un modelo táctico de Planificación de Adquisición de Bulbos (PAB) que minimiza los costos de adquisición, transporte e inventario del material vegetal. El modelo incorpora y explicita conceptos esenciales en el proceso de toma de decisiones como: sustituibilidad, precocidad, mantención y costo de inventario, diversificación de riesgo, centros de origen y estacionalidad de la disponibilidad del material vegetal.  El modelo PAB fue programado con datos provenientes de la industria y comparado con la programación real realizada por una empresa colaboradora para la temporada 2002. De los resultados, es posible indicar que el mayor impacto sobre los costos se observó en cuanto a la programación de embarques y la mantención de inventario.
	
En la literatura no se ha abordado directamente este problema, por lo que el modelo de programación matemática y la solución presentada en este paper, pueden ser consideradas como un trabajo precursor en el área de planificación agregada de la producción en empresas dedicadas a la floricultura.

 El modelo PAB presentó una buena acogida por parte de los ejecutivos de la empresa colaboradora, los cuales están considerando su incorporación para la planificación de la adquisición de bulbos de la próxima temporada.

\section{Acknowlegments}

Los autores agradecen en forma especial los comentarios aportados por William Foster y Rob A.C.M. Broekmeulen, quienes contribuyeron a mejorar el contenido y la presentación original de este trabajo. También, se agradece el soporte logístico y los datos provistos por  la empresa Pacific Flowers. Por último, se agradece la beca otorgada por la Comisión Nacional de Investigación Científica y Tecnológica de Chile (CONICYT), por permitir financiar parte de esta investigación.



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